Soft close toilet seat issue

If the problem is mineral clogging of the under-rim channels on the toilet, and this does produce the symptoms you describe, then neither CLR no vinegar may be enough to clear up the blockage. The traditional treatment is a cup or two of muriatic acid (HCl 36%, from the hardware store) down the overflow hole in the tank. That goes directly to the rim vents. Pour the stuff in, wait an hour, and flush. Gloves and eye protection are recommmended, and a funnel so as to avoid spillage and corroding the metal parts in your tank. This looks like decent writeup of the method: How to Fix a Slow Toilet

In the US, the typical toilet rough in size is 12", which I believe goes from the toilet bolts (center of the drain) to the studs (so subtract 1/2" for drywall). You can also find toilets made for 10" and 14" rough ins. But, to get more clearance, the easiest thing to do is swap out the style of toilet from an elongated bowl to a round bowl. If you're a man visiting the library, it means you have less space to hang free, but it gives you a lot more space to get around the bathroom.

To answer your first question, you can move the drain, but it involves taking a jackhammer to the concrete. Not a fun project if you don't like messes.